and assuming that the spatial variation of the electric field is much longer than the Compton wavelength \(\lambda \gg \lambda _c\) , it is possible to describe the Schwinger effect at any point \(z_\text{fix}\)  independently. Our goal is to use the analytical solution of the one-particle distribution function \(F(\mathbf {p},t)\)  for a homogeneous electric field [1]}, [2]}, [3]}. Thus, we approximate the current density \(v_z(\mathbf {p},t,z)\)  as
\(v_z(\mathbf {p},t,z)\approx v_{z}^{h}\Big (\mathbf {p},t,E_0f(z)\Big )\) 
